One and Two-STAGE BINOMIAL OPTION PRICING MODEL - Method...

BARUCH COLLEGE ndash DEPARTMENT OF ECONOMICS amp FINANCE ndash FIN4710

Professor Chris Droussiotis

1

LECTURE 9

One and Two-STAGE BINOMIAL OPTION PRICING MODEL - Method 2

INPUT OUTPUT

Example I - Single Stage (Call Option) PERIOD 0 PERIOD 1

S = 4500$

u = 110x Su= 4950 Cu= 950

d = 090x

X = 4000$ S = 4500 690

i = 500

Freq= 1 Sd = 4050 Cd= 050

Stages= 1

p = 075

1-p= 025

C= 690

Example II (Call option w no Dividends) PERIOD 0 PERIOD 1 PERIOD 2

Su^2= 5445 Cu^2= 1445

S = 4500$

u = 110x Su= 4950 C1= 1140

d = 090x

X = 4000$ S = 4500 892 4455 Cud= 455

i = 500

Freq= 1 Sd = 4050 C2= 325

Stages= 2

p = 075 Sd^2= 3645 Cd^2= 000

1-p= 025

C= 892

Example III (Put Option w no Dividends) PERIOD 0 PERIOD 1 PERIOD 2

Su^2= 7502 Pu^2= 000

S = 6200$

u = 110x Su= 6820 064

d = 095x

X = 7000$ S = 6200 125 6479 Pud= 521

i = 800

Freq= 1 Sd = 5890 591

Stages= 2

p = 087 Sd^2= 5596 Pd^2= 1405

1-p= 013

P= 125

Example IV (Call Option w Dividends) PERIOD 0 PERIOD 1 PERIOD 1(x-div) PERIOD 2

Su^2= 3729 Cu^2= 1229

x-dividend

S = 3000$ Su= 3450 3243 862

u = 115x

d = 090x S = 3000 584 2919 Cud= 419

X = 2500$ 2919

i = 500 Sd = 2700 2538 239

Div (δ)= 600

Sd^2= 2284 Cd^2= 000

Freq= 1 p = 060

Stages= 2 1-p= 040

C= 584



2

BLACK-SHOLES OPTION PRICING MODEL

The BlackndashScholes model is a mathematical description of financial markets and

derivative investment instruments The model develops partial differential equations

whose solution the BlackndashScholes formula is widely used in the pricing of European-

style options

Black-Scholes Model - Definition

A mathematical formula designed to price an option as a function of certain variables-

generally stock price striking price volatility time to expiration dividends to be paid

and the current risk-free interest rate

Black-Scholes Model - Introduction

The Black-Scholes model is a tool for equity options pricing Prior to the development of

the Black-Scholes Model there was no standard options pricing method and nobody can

put a fair price to charge for options The Black-Scholes Model turned that guessing

game into a mathematical science which helped develop the options market into the

lucrative industry it is today Options traders compare the prevailing option price in the

exchange against the theoretical value derived by the Black-Scholes Model in order to

determine if a particular option contract is over or under valued hence assisting them in

their options trading decision The Black-Scholes Model was originally created for the



3

pricing and hedging of European Call and Put options as the American Options market

the CBOE started only 1 month before the creation of the Black-Scholes Model The

difference in the pricing of European options and American options is that options

pricing of European options do not take into consideration the possibility of early

exercising American options therefore command a higher price than European options

due to the flexibility to exercise the option at anytime The classic Black-Scholes Model

does not take this extra value into consideration in its calculations

Black-Scholes Model Assumptions

There are several assumptions underlying the Black-Scholes model of calculating options

pricing The most significant is that volatility a measure of how much a stock can be

expected to move in the near-term is a constant over time The Black-Scholes model also

assumes stocks move in a manner referred to as a random walk at any given moment

they are as likely to move up as they are to move down These assumptions are combined

with the principle that options pricing should provide no immediate gain to either seller

or buyer

The exact 6 assumptions of the Black-Scholes Model are

1 Stock pays no dividends

2 Option can only be exercised upon expiration

3 Market direction cannot be predicted hence Random Walk

4 No commissions are charged in the transaction

5 Interest rates remain constant

6 Stock returns are normally distributed thus volatility is constant over time

As you can see the validity of many of these assumptions used by the Black-Scholes

Model is questionable or invalid resulting in theoretical values which are not always

accurate Hence theoretical values derived from the Black-Scholes Model are only good

as a guide for relative comparison and is not an exact indication to the over or under

priced nature of a stock option



4

Model assumptions

The BlackndashScholes model of the market for a particular equity makes the following

explicit assumptions

It is possible to borrow and lend cash at a known constant risk-free interest rate

This restriction has been removed in later extensions of the model

The price follows a Geometric Brownian motion with constant drift and volatility

It follows from this that the return is a Log-normal distribution This often implies

the validity of the efficient-market hypothesis

There are no transaction costs or taxes

The stock does not pay a dividend (see below for extensions to handle dividend

payments)

All securities are perfectly divisible (ie it is possible to buy any fraction of a

share)

There are no restrictions on short selling

There is no arbitrage opportunity

Options use the European exercise terms which dictate that options may only be

exercised on the day of expiration

From these conditions in the market for an equity (and for an option on the equity) the

authors show that it is possible to create a hedged position consisting of a long position

in the stock and a short position in [calls on the same stock] whose value will not depend

on the price of the stock[3]

Several of these assumptions of the original model have been removed in subsequent

extensions of the model Modern versions account for changing interest rates (Merton

1976) transaction costs and taxes (Ingerson 1976) and dividend payout (Merton 1973)

The Black Scholes formula calculates the price of European put and call options It can

be obtained by solving the BlackndashScholes partial differential equation

The value of a call option in terms of the BlackndashScholes parameters is

C (St) = SN (d1) ndash Xe ndashr(T-t)

N(d2)

d1 = [ ln (SoX) + (r + σ2 2) (T ndash t) ] [ σ SQR of (T ndash t) ]

d2 = d1 ndash σ SQRT of (T ndash t)



5

The price of a put option is

P (S t) = Xe ndashr(T-t) ndash S+(SN(d1) ndash Xe ndashr(T-t) N(d2)) = Xe ndashr(T-t) ndash S+C (St)

For both as above

N(bull) is the cumulative distribution function of the standard normal distribution

T - t is the time to maturity

S is the spot price of the underlying asset

X is the strike price

r is the risk free rate (annual rate expressed in terms of continuous compounding)

σ is the volatility in the log-returns of the underlying

Interpretation

N(d1) and N(d2) are the probabilities of the option expiring in-the-money under the

equivalent exponential martingale probability measure (numeacuteraire = stock) and the

equivalent martingale probability measure (numeacuteraire = risk free asset) respectively The

equivalent martingale probability measure is also called the risk-neutral probability

measure Note that both of these are probabilities in a measure theoretic sense and

neither of these is the true probability of expiring in-the-money under the real probability

measure In order to calculate the probability under the real (physical) probability

measure additional information is require

dmdashthe drift term in the physical measure or equivalently the market price of risk

Example

Suppose you want to value a call option under the following circumstances

Stock Price S0 = 100

Exercise Price X=95

Interest Rate r= 10

Dividend Yield δ = 0

Time to expiration T = 25 (one-quarter year)

Standard Deviation σ = 50

First calculate



6

d1 = [ln (10095) + (10-0 + 522)25] [ 5 SQRT of 25] = 43

d2 = 43 - 5 SQRT of 25 = 18

Next find N (d1) and d N(d2) The normal distribution function is tabulated and may be

found in many statistics books A table of N (d) is provided as Table 162 in the book

page 521 The normal distribution function N(d) is also provided in any spreadsheet

program In Excel the function name is NORMSDIST so using EXCEL (using

interpolation for 43) we find that

N(43) = 6664

N(18) = 5714

Finally remember that with dividends (δ) = o

S0 e ndash δT

= S0

Thus the value of the call option is

C = 100 x 6664 ndash 95e -10x025

x 5714

=6664 ndash 5294 = $1370



7

BLACK-SCHOLES OPTION VALUATION METHOD BS - CALL OPTION

A B C D E F G Compound at e

5

6 INPUT OUTPUT Face Value 100$

7 Interest 10

8 Standard Deviation (σ) = 05 d1 = 0430 Years 109 Expiration (in years) (T) = 025 d2 = 0180

10 Risk-Free Rate (Annual) (i) = 01 N(d1) = 0666 Description Compound FV

11 Stock Price (S ) = 100 N(d2) = 0571 Annual 1 25937425 12 Exercise Price (X) = 95 Semi 2 26532977

13 Dividend Yield (annual) (δ) = 0 C = 136953 Quarterly 4 26850638 Monthly 12 27070415

Daily 365 27179096

LONG CALCULATION (Break Down Approach) Hourly 8760 27182663

D1 = Ln ( S X ) ( i-δ+σ^2 2 ) σradict By Minute 525600 27182816 D1 = 0051293294 005625 025 By Second 31536000 27182819

Infinite e 27182818

D1 = 043017

N (d1) = 066647

PV calculation using e

D2= 018017 e = PV x (1+i)^t

N (d2) = 057149 PV = e (1+i)^t

PV = e ^-itC = 1370

2 BLACK-SCHOLES OPTION VALUATION METHOD BS - PUT OPTION


32


34 Interest 10

35 Standard Deviation (σ) = 05 d1 = 0430 Years 10

36 Expiration (in years) (T) = 025 d2 = 0180


38 Stock Price (S ) = 100 N(d2) = 0571 Annual 1 25937425

39 Exercise Price (X) = 95 Semi 2 26532977

40 Dividend Yield (annual) (δ) = 0 P = 63497 Quarterly 4 26850638

Monthly 12 27070415

Daily 365 27179096


D1 = Ln ( S X ) ( i-δ+σ^2 2 ) σradict By Minute 525600 27182816

D1 = 0051293294 005625 025 By Second 31536000 27182819

Infinite e 27182818

D1 = 0430173178

N (d1) = 0666465164


D2= 0180173178 e = PV x (1+i)^t

N (d2) = 0571491692 PV = e (1+i)^t

PV = e ^-itP = 63497

3 PUT-CALL PARITY METHOD FOR CALCULATING THE PUT OPTION KNOWING THE CALL PRICE (same data as above)

C - P = S - X e -it

P = Xe -it - S + C

P = 63497



8

Review ndash Options

C = S e -δT

N (d1) ndash X e ndashiT

N (d2)

P = X e ndashiT

(1- N (d2)) ndash S e -δT

(1 - N (d1))

Volatility is the question on the BS ndashwhich assumes constant SD throughout the exercise

period - The time series of implied volatility

THE PUT ndash CALL PARITY RELATIONSHIP

Put prices can be derived simply from the prices of call

European Put or Call options are linked together in an equation known as the Put-

Call parity relationship

St lt= X St gt X

Payoff of Call Held 0 St - X

Payoff of Put Written -(X ndash St) 0

Total St ndash X St ndash X

PV (x) = X e ndashrt

The option has a payoff identical to that of the leveraged equity position the costs of

establishing them must be equal

C ndash P Cost of Call purchased = Premium received from Put written

The leverage Equity position requires a net cash outlay of S ndash X e ndashrt

the Cost

of the stock less the process from borrowing

C ndash P = S ndash X e ndashrt

PUT-CALL Parity Relationship - proper relationship

between Call and Put

Example 163



9

S = $110

C = $14 for 6 months with X = $105

P = $5 for 6 months with X=$105

rf = 50 (continuously compounding at e )

Assumptions

C ndash P = S ndash X e ndashrT

14 ndash 5 = 110 ndash 105 e ndash 05 x 05

9 = 759

This a violation of parityhellip Indicates mispricing and leads to Arbitrage

Opportunity

You can buy relatively cheap portfolio (buy the stock plus borrowing position

represented on the right side of the equation and sell the expensive portfolio

STRATEGY ndash In six months the stock will be worth Sr so you borrow PV of X

($105) and pay back the loan with interest resulting in cash outflow of $105

Sr ndash 105 writing the call if Sr exceeds 105

Purchase Puts will pay 105 ndash Sr if the stock is below the $105

Strategy Immediate

CF

CF if

Sr lt 105

CF if

Sr gt 105

1 Buy Stock -11000 Sr Sr

2 Borrow Xe ndashiT

= $10241 +10241 -105 -105

3 Sell Call 1400 0 -(Sr ndash 105)

4 Buy Call -500 105 ndash Sr 0

141 0 0

Whish is the difference of between 900 and 759 ndash riskless return

This applies if No dividends and under the European option

If Dividend then



10

P = C ndash S + PV (X) + PV ( Dividend) hellip Representing that the Dividend (δ) is

paid during the life of the option

Example

Using the IBM example ndash today is February 6

X = $100 (March calls)

T = 42 days

C = $280

P = $647

S = 9614

I = 20

δ = 0

P = C ndash S + PV (X) + PV ( Dividend) or P = C + PV (X) ndash S + PV (δ)

647 = 280 + 100 (1+002)42365

- 9614 + 0

647 = 663 is not that valuable to go after the reprising arbitrage

PUT OPTION VALUATION

P = X e ndashiT


(1 - N (d1))

Using the data from previous example

P = 95 e ndash 10x25

(1 ndash 005714) ndash 100 (1 ndash 06664)

P = 635

PUT-CALL Parity

P = C + PV (X) ndash So + PV (Div)

P = 1370 + 95 e -10 X 025

ndash 100 + 0

Hedge Ratios amp the BS format



11

The Hedge ratio is commonly called the Option Delta Is the change in the price

of call option for $1 increased in the stock price

This is the slope of value function evaluated at the current stock price

For Example

Slope of the curve at S = $120 equals 60 As the stock increases by $1 the option

increase on 060

For every Call Option Written 60 shares of stock would be needed to hedge the

Investment portfolio

For example if one writes 10 options and holds 6 shares of stock

H = 60 helliphelliphellip a $1 increase in stock will result $6 gain ($1x 6 shares) and with

the loss of $6 on 10 options written (10 x $060)

The Hedge Ratio for a Call is N (d1)

with the hedge ratio for a Put [N (d1) ndash 1]

N (d) is the area under standard deviation (normal)

Therefore the Call option Hedge Ratio must be positive and less than 10

And the Put option Hedge Ratio is negative and less than 10

Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM

Which portfolio has a greater dollar exposure to IBM price movement

Using the Hedge ratio you could answer that question

Each Option change in value by H dollars for each $1 change in stock price



12

If H = 06 then 750 options = equivalent 450 shares (06 x 750)

Portfolio A = 450 equivalent + 200 shares which is less than Portfolio B with 800

shares



2

BLACK-SHOLES OPTION PRICING MODEL

The BlackndashScholes model is a mathematical description of financial markets and

derivative investment instruments The model develops partial differential equations

whose solution the BlackndashScholes formula is widely used in the pricing of European-

style options

Black-Scholes Model - Definition

A mathematical formula designed to price an option as a function of certain variables-

generally stock price striking price volatility time to expiration dividends to be paid

and the current risk-free interest rate

Black-Scholes Model - Introduction

The Black-Scholes model is a tool for equity options pricing Prior to the development of

the Black-Scholes Model there was no standard options pricing method and nobody can

put a fair price to charge for options The Black-Scholes Model turned that guessing

game into a mathematical science which helped develop the options market into the

lucrative industry it is today Options traders compare the prevailing option price in the

exchange against the theoretical value derived by the Black-Scholes Model in order to

determine if a particular option contract is over or under valued hence assisting them in

their options trading decision The Black-Scholes Model was originally created for the



3















or buyer















4

Model assumptions










payments)


share)
















N(d2)





5



For both as above







Interpretation










Example



Exercise Price X=95

Interest Rate r= 10




First calculate



6

d1 = [ln (10095) + (10-0 + 522)25] [ 5 SQRT of 25] = 43

d2 = 43 - 5 SQRT of 25 = 18






N(43) = 6664

N(18) = 5714


S0 e ndash δT

= S0


C = 100 x 6664 ndash 95e -10x025

x 5714

=6664 ndash 5294 = $1370



7



5


7 Interest 10





Daily 365 27179096



Infinite e 27182818

D1 = 043017

N (d1) = 066647


D2= 018017 e = PV x (1+i)^t

N (d2) = 057149 PV = e (1+i)^t

PV = e ^-itC = 1370



32


34 Interest 10







Monthly 12 27070415

Daily 365 27179096



D1 = 0051293294 005625 025 By Second 31536000 27182819

Infinite e 27182818

D1 = 0430173178

N (d1) = 0666465164


D2= 0180173178 e = PV x (1+i)^t

N (d2) = 0571491692 PV = e (1+i)^t

PV = e ^-itP = 63497


C - P = S - X e -it

P = Xe -it - S + C

P = 63497



8


C = S e -δT


N (d2)

P = X e ndashiT


(1 - N (d1))







St lt= X St gt X









the Cost





Example 163



9

S = $110




Assumptions



9 = 759


Opportunity







Strategy Immediate

CF

CF if

Sr lt 105

CF if

Sr gt 105


2 Borrow Xe ndashiT

= $10241 +10241 -105 -105



141 0 0



If Dividend then



10



Example



T = 42 days

C = $280

P = $647

S = 9614

I = 20

δ = 0


647 = 280 + 100 (1+002)42365

- 9614 + 0



P = X e ndashiT


(1 - N (d1))




P = 635

PUT-CALL Parity


P = 1370 + 95 e -10 X 025

ndash 100 + 0




11




For Example


increase on 060











Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM






12



shares



3















or buyer















4

Model assumptions










payments)


share)
















N(d2)





5



For both as above







Interpretation










Example



Exercise Price X=95

Interest Rate r= 10




First calculate



6

d1 = [ln (10095) + (10-0 + 522)25] [ 5 SQRT of 25] = 43

d2 = 43 - 5 SQRT of 25 = 18






N(43) = 6664

N(18) = 5714


S0 e ndash δT

= S0


C = 100 x 6664 ndash 95e -10x025

x 5714

=6664 ndash 5294 = $1370



7



5


7 Interest 10





Daily 365 27179096



Infinite e 27182818

D1 = 043017

N (d1) = 066647


D2= 018017 e = PV x (1+i)^t

N (d2) = 057149 PV = e (1+i)^t

PV = e ^-itC = 1370



32


34 Interest 10







Monthly 12 27070415

Daily 365 27179096



D1 = 0051293294 005625 025 By Second 31536000 27182819

Infinite e 27182818

D1 = 0430173178

N (d1) = 0666465164


D2= 0180173178 e = PV x (1+i)^t

N (d2) = 0571491692 PV = e (1+i)^t

PV = e ^-itP = 63497


C - P = S - X e -it

P = Xe -it - S + C

P = 63497



8


C = S e -δT


N (d2)

P = X e ndashiT


(1 - N (d1))







St lt= X St gt X









the Cost





Example 163



9

S = $110




Assumptions



9 = 759


Opportunity







Strategy Immediate

CF

CF if

Sr lt 105

CF if

Sr gt 105


2 Borrow Xe ndashiT

= $10241 +10241 -105 -105



141 0 0



If Dividend then



10



Example



T = 42 days

C = $280

P = $647

S = 9614

I = 20

δ = 0


647 = 280 + 100 (1+002)42365

- 9614 + 0



P = X e ndashiT


(1 - N (d1))




P = 635

PUT-CALL Parity


P = 1370 + 95 e -10 X 025

ndash 100 + 0




11




For Example


increase on 060











Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM






12



shares



4

Model assumptions










payments)


share)
















N(d2)





5



For both as above







Interpretation










Example



Exercise Price X=95

Interest Rate r= 10




First calculate



6

d1 = [ln (10095) + (10-0 + 522)25] [ 5 SQRT of 25] = 43

d2 = 43 - 5 SQRT of 25 = 18






N(43) = 6664

N(18) = 5714


S0 e ndash δT

= S0


C = 100 x 6664 ndash 95e -10x025

x 5714

=6664 ndash 5294 = $1370



7



5


7 Interest 10





Daily 365 27179096



Infinite e 27182818

D1 = 043017

N (d1) = 066647


D2= 018017 e = PV x (1+i)^t

N (d2) = 057149 PV = e (1+i)^t

PV = e ^-itC = 1370



32


34 Interest 10







Monthly 12 27070415

Daily 365 27179096



D1 = 0051293294 005625 025 By Second 31536000 27182819

Infinite e 27182818

D1 = 0430173178

N (d1) = 0666465164


D2= 0180173178 e = PV x (1+i)^t

N (d2) = 0571491692 PV = e (1+i)^t

PV = e ^-itP = 63497


C - P = S - X e -it

P = Xe -it - S + C

P = 63497



8


C = S e -δT


N (d2)

P = X e ndashiT


(1 - N (d1))







St lt= X St gt X









the Cost





Example 163



9

S = $110




Assumptions



9 = 759


Opportunity







Strategy Immediate

CF

CF if

Sr lt 105

CF if

Sr gt 105


2 Borrow Xe ndashiT

= $10241 +10241 -105 -105



141 0 0



If Dividend then



10



Example



T = 42 days

C = $280

P = $647

S = 9614

I = 20

δ = 0


647 = 280 + 100 (1+002)42365

- 9614 + 0



P = X e ndashiT


(1 - N (d1))




P = 635

PUT-CALL Parity


P = 1370 + 95 e -10 X 025

ndash 100 + 0




11




For Example


increase on 060











Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM






12



shares



5



For both as above







Interpretation










Example



Exercise Price X=95

Interest Rate r= 10




First calculate



6

d1 = [ln (10095) + (10-0 + 522)25] [ 5 SQRT of 25] = 43

d2 = 43 - 5 SQRT of 25 = 18






N(43) = 6664

N(18) = 5714


S0 e ndash δT

= S0


C = 100 x 6664 ndash 95e -10x025

x 5714

=6664 ndash 5294 = $1370



7



5


7 Interest 10





Daily 365 27179096



Infinite e 27182818

D1 = 043017

N (d1) = 066647


D2= 018017 e = PV x (1+i)^t

N (d2) = 057149 PV = e (1+i)^t

PV = e ^-itC = 1370



32


34 Interest 10







Monthly 12 27070415

Daily 365 27179096



D1 = 0051293294 005625 025 By Second 31536000 27182819

Infinite e 27182818

D1 = 0430173178

N (d1) = 0666465164


D2= 0180173178 e = PV x (1+i)^t

N (d2) = 0571491692 PV = e (1+i)^t

PV = e ^-itP = 63497


C - P = S - X e -it

P = Xe -it - S + C

P = 63497



8


C = S e -δT


N (d2)

P = X e ndashiT


(1 - N (d1))







St lt= X St gt X









the Cost





Example 163



9

S = $110




Assumptions



9 = 759


Opportunity







Strategy Immediate

CF

CF if

Sr lt 105

CF if

Sr gt 105


2 Borrow Xe ndashiT

= $10241 +10241 -105 -105



141 0 0



If Dividend then



10



Example



T = 42 days

C = $280

P = $647

S = 9614

I = 20

δ = 0


647 = 280 + 100 (1+002)42365

- 9614 + 0



P = X e ndashiT


(1 - N (d1))




P = 635

PUT-CALL Parity


P = 1370 + 95 e -10 X 025

ndash 100 + 0




11




For Example


increase on 060











Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM






12



shares



6

d1 = [ln (10095) + (10-0 + 522)25] [ 5 SQRT of 25] = 43

d2 = 43 - 5 SQRT of 25 = 18






N(43) = 6664

N(18) = 5714


S0 e ndash δT

= S0


C = 100 x 6664 ndash 95e -10x025

x 5714

=6664 ndash 5294 = $1370



7



5


7 Interest 10





Daily 365 27179096



Infinite e 27182818

D1 = 043017

N (d1) = 066647


D2= 018017 e = PV x (1+i)^t

N (d2) = 057149 PV = e (1+i)^t

PV = e ^-itC = 1370



32


34 Interest 10







Monthly 12 27070415

Daily 365 27179096



D1 = 0051293294 005625 025 By Second 31536000 27182819

Infinite e 27182818

D1 = 0430173178

N (d1) = 0666465164


D2= 0180173178 e = PV x (1+i)^t

N (d2) = 0571491692 PV = e (1+i)^t

PV = e ^-itP = 63497


C - P = S - X e -it

P = Xe -it - S + C

P = 63497



8


C = S e -δT


N (d2)

P = X e ndashiT


(1 - N (d1))







St lt= X St gt X









the Cost





Example 163



9

S = $110




Assumptions



9 = 759


Opportunity







Strategy Immediate

CF

CF if

Sr lt 105

CF if

Sr gt 105


2 Borrow Xe ndashiT

= $10241 +10241 -105 -105



141 0 0



If Dividend then



10



Example



T = 42 days

C = $280

P = $647

S = 9614

I = 20

δ = 0


647 = 280 + 100 (1+002)42365

- 9614 + 0



P = X e ndashiT


(1 - N (d1))




P = 635

PUT-CALL Parity


P = 1370 + 95 e -10 X 025

ndash 100 + 0




11




For Example


increase on 060











Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM






12



shares



7



5


7 Interest 10





Daily 365 27179096



Infinite e 27182818

D1 = 043017

N (d1) = 066647


D2= 018017 e = PV x (1+i)^t

N (d2) = 057149 PV = e (1+i)^t

PV = e ^-itC = 1370



32


34 Interest 10







Monthly 12 27070415

Daily 365 27179096



D1 = 0051293294 005625 025 By Second 31536000 27182819

Infinite e 27182818

D1 = 0430173178

N (d1) = 0666465164


D2= 0180173178 e = PV x (1+i)^t

N (d2) = 0571491692 PV = e (1+i)^t

PV = e ^-itP = 63497


C - P = S - X e -it

P = Xe -it - S + C

P = 63497



8


C = S e -δT


N (d2)

P = X e ndashiT


(1 - N (d1))







St lt= X St gt X









the Cost





Example 163



9

S = $110




Assumptions



9 = 759


Opportunity







Strategy Immediate

CF

CF if

Sr lt 105

CF if

Sr gt 105


2 Borrow Xe ndashiT

= $10241 +10241 -105 -105



141 0 0



If Dividend then



10



Example



T = 42 days

C = $280

P = $647

S = 9614

I = 20

δ = 0


647 = 280 + 100 (1+002)42365

- 9614 + 0



P = X e ndashiT


(1 - N (d1))




P = 635

PUT-CALL Parity


P = 1370 + 95 e -10 X 025

ndash 100 + 0




11




For Example


increase on 060











Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM






12



shares



8


C = S e -δT


N (d2)

P = X e ndashiT


(1 - N (d1))







St lt= X St gt X









the Cost





Example 163



9

S = $110




Assumptions



9 = 759


Opportunity







Strategy Immediate

CF

CF if

Sr lt 105

CF if

Sr gt 105


2 Borrow Xe ndashiT

= $10241 +10241 -105 -105



141 0 0



If Dividend then



10



Example



T = 42 days

C = $280

P = $647

S = 9614

I = 20

δ = 0


647 = 280 + 100 (1+002)42365

- 9614 + 0



P = X e ndashiT


(1 - N (d1))




P = 635

PUT-CALL Parity


P = 1370 + 95 e -10 X 025

ndash 100 + 0




11




For Example


increase on 060











Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM






12



shares



9

S = $110




Assumptions



9 = 759


Opportunity







Strategy Immediate

CF

CF if

Sr lt 105

CF if

Sr gt 105


2 Borrow Xe ndashiT

= $10241 +10241 -105 -105



141 0 0



If Dividend then



10



Example



T = 42 days

C = $280

P = $647

S = 9614

I = 20

δ = 0


647 = 280 + 100 (1+002)42365

- 9614 + 0



P = X e ndashiT


(1 - N (d1))




P = 635

PUT-CALL Parity


P = 1370 + 95 e -10 X 025

ndash 100 + 0




11




For Example


increase on 060











Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM






12



shares



10



Example



T = 42 days

C = $280

P = $647

S = 9614

I = 20

δ = 0


647 = 280 + 100 (1+002)42365

- 9614 + 0



P = X e ndashiT


(1 - N (d1))




P = 635

PUT-CALL Parity


P = 1370 + 95 e -10 X 025

ndash 100 + 0




11




For Example


increase on 060











Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM






12



shares



11




For Example


increase on 060











Example 165

2 Portfolios

Portfolio A B

BUY 750 IBM Calls

200 Shares of IBM

800 shares of IBM






12



shares



12



shares

One and Two-STAGE BINOMIAL OPTION PRICING MODEL - Method...

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Transcript of One and Two-STAGE BINOMIAL OPTION PRICING MODEL - Method...